Sensitivity function for various geoelectric arrays F. Chitea1, Georgescu P 1

1 University of Bucharest, Faculty of Geology and Geophysics

Resistivity measurements are commonly used for investigating various types of environmental problems. As a general principle, the acquisition techniques consist in measuring, between two electrodes (M, N), the difference of potential generated by the electrical current which is injected into the ground via two other electrodes (the transmitting line AB). Various type of electrode combinations (A,B,M,N) can be used when investigating a particular area (Schlumberger, Wenner –α, Wenner –β, Wenner - γ, pole-pole, pole-dipole, dipole-dipole, gradient), but the selection of the proper geoelectric array to be used for a specific investigation has to be performed by considering the estimated depth of the target and its characteristics, the signal strength, the presence or absence of noise sources, the main expected variation in the electrical resistivity of the geological medium (i.e. along the vertical, along the horizontal, or in both directions), and the sensitivity of the array.

As expected, the sensitivity function of geoelectric arrays depends on the relative positions of the electrodes. In terms of resolution and of depth of investigation, each of the above mentioned arrays has its own specific benefits and limitations. Moreover, for certain arrays, the field set-up is another element to be carefully considered, in order to avoid violating the resistivity method basic principles.

Sensitivity can be described by the Frechet derivative. Calculating the sensitivity consists in determining the alteration that a resistivity variation, generated within a small volume of soil, induces in the potential drop measured between the MN electrodes. The higher the value of the sensitivity function, the larger is the recorded influence due to the resistivity change applied to the small volume of soil (Loke, 2004).

The Frechet derivative, obtained by considering the injection of the electrical current by means of just one non-linear injection electrode, and by also taking into account only one receiving electrode, is given by equation (1)(Loke, 2004):

[ ] [ ] 5.12225.1222

22

23D)(

)(4

1 z)y,(x, Fzyaxzyx

zyaxx++−++

++−⋅=

π (1)

a- electrodes separation (x,y,z ) indicates the location of the small volume being subject to the resistivity change (as shown in Fig. 1)

Fig. 1. 2D view of the soil layered model, indicating the configuration of the electrodes

and the volume for which the sensitivity function is calculated.

For quadripolar arrays, a summing accounting for the contribution of the other intervening electrodes has to be performed, in order to obtain the overall sensitivity.

By studying the 1D distribution of the sensitivity function, Loke has shown that the Wenner array has a better vertical resolution than the pole-pole array.

In the following, 2D variations in the sensitivity function shall be discussed, specifically addressing, for different arrays, their lateral and vertical resolutions. Generally speaking, the contour pattern of the represented sensitivity function is different for the various arrays. However, a common feature of all the investigated quadripolar and tripolar array types is the systematic occurrence of the highest sensitivity values in the immediate vicinity of the electrodes.

-1000-900-800-700-600-500-400-300-200-100010020030040050060070080090010001100

x10

-1600-1400-1200-1000-800-600-400-200020040060080010001200140016001800

x10

Fig.2. Sensitivity distributions for Wenner and Dipole-Dipole acquisition systems Positive values indicate the regions where the presence of a body which is more resistive

than its background will result in an enhancement of the measured resistivity, while in the areas with negative sensitivities, the same body will induce a reduction of the measured resistivity.

By analyzing the sensitivity function for various geoelectric arrays (Wenner,

Schlumberger, Dipole-Dipole), noticeable advantages and limitation have been identified. In the following, we will underline only recommendations to be considered in choosing an array best-suited to site-specific characteristics:

- Wenner array has the strongest signal strength, and this feature recommends the use of this particular electrode configuration when the survey is carried out in the presence of geoelectrical noise.

- Dipole-dipole array is sensitive when a lateral resistivity variation is encountered, and it is therefore the best choice when mapping vertical structures.

- Pole-pole has the deepest depth of investigation: it is therefore very practical when the target is located at large depth and displays a resistivity which is highly contrasting with that of the background.

Not only the benefits, but also shortcomings related to the utilization a specific array can

be deduced by analyzing the sensitivity function. As an example, the pole-pole array has the widest horizontal coverage and the deepest depth of investigation, but it can pick up a large amount of telluric noise and has the poorest resolution (Looke, 2004).

When multi-electrode measuring systems are available, one can use hybrid electrode configurations, in order to minimize the specific weakness of each individual array. Widely used

hybrid configurations are Wenner-Schlumberger and Gradient-Dipole, highly recommended when complex resistivity distributions are expected (Chitea et al., 2009).

Fig. 3 – Resistivity pseudosections obtained on the synthetic model using:

a) Wenner-Schlumberger array b) Dipole-Gradient array Results obtained on the synthetic model by using hybrid acquisition techniques (Wenner-Schlumberger and Gradient-Dipole) are presented as resistivity pseudosections in Fig. 3. By comparing the resistivity pseudosections displayed in figure 3, which were obtained by using different acquisition arrays, there can be noticed that the Dipole-Gradient array secures a better coverage of the in-depth investigation. In areas where both types of geological structures (vertical and horizontal) are expected to occur, choosing between Wenner and the dipole-dipole array will severely decrease the quality of the results. Under such circumstances, better results might be obtained by using the Wenner-Schlumberger array. The signal strength of this hybrid array is weaker than the one provided by the Wenner array, but the depth of investigation is increased and lateral coverage is improved when using Wenner-Schlumberger, again by comparison with Wenner. As compared to the dipole-dipole array, there can be noticed that the signal strength of the Wenner-Schlumberger is higher, but the laterally covered area is smaller. By comparing the sensitivity function for various arrays, specific disadvantages can be outlined, and there can be also issued recommendations for using one set-up or another, such an analysis being recommended before the beginning of the acquisition campaign. Electrodes separation is another important issue which has to be considered when geoelectric measurements are designed. When the depth of the target is large, the length of the transmission line (AB) is extended, in order to reach the desired investigation depth. Recent studies have shown great interest for shallower targets as well. The sensitivity function shows that for smaller electrode separations, the resolution in the vicinity of the surface is increased, but some other important aspects must be alternatively considered, when smaller electrodes separations are required. Electrical prospecting method uses metallic electrodes which are inserted into the ground up to a depth which depends on the field conditions and on the type of array. Consequently, current electrodes, as well as potential electrodes, can only approximately be described as point-electrodes. That circumstance may induce certain difficulties in the interpretation of the resulted geological data, for which the usual approach relies on theoretical models devised under the assumption of point sources/sinks of current (Geprgescu P., 1982). In order to illustrate the difference between values computed under the assumption of a point-electrode, and values obtained under the more realistic assumption that a real electrode, of finite length, was utilized, there has been computed the potential generated in a certain point M , by a current source of

length , that injected a current of intensity a I into a homogeneous and isotropic half-space of resistivity ρ . Assuming that the current distribution was uniform along the electrode length , the corresponding expression is:

a

aaraar

aIV

−+

++=

22

22

ln4

.πρ (2)

When using a Schlumberger array, the apparent resistivity corresponding to a point-electrode would be equal to the resistivity of the considered half-space, while the apparent resistivity corresponding to the line-electrode is:

22

.ar

ra

+=

ρρ (3)

The ratio between the apparent resistivity corresponding to the line-electrode and the apparent resistivity corresponding to a point-electrode is:

F

00.10.20.30.40.50.60.70.80.9

1

0 1 2 3 4 5 6 7 8 9 10

x

F F

12 +=

ξξF (4)

where ar

=ξ

Fig. 4. Variation of function F

The diagram illustrating the function variation (fig.4) indicates that the apparent resistivity computed in the case of a line-electrode is equal to the apparent resistivity corresponding to a point-electrode only if the distance between the current source and the point

F

M (where the electrical potential is measured) is much larger that the electrode burying-depth a . The distance r between the current source and the point M has to be at least four times larger than the electrode burying-depth, in order to secure differences smaller than 5% between the actually computed resistivity values and values existing in the ideal case. In addition, by taking into account the circumstance that both the current electrodes, and the potential electrodes, are actually linear, the electrodes separation must be at least 5-7 times larger than their burying-depth. The latter observation is of utmost importance in carrying out resistivity tomographies. One has to select an optimum value for the ratio of electrodes-separation versus burying-depth, in order to secure appropriate results, especially when taking into account the circumstance that within the various electrodes combinations utilized by the multi-electrode measuring systems, electrodes separations are variable. References: Chitea F., Ioane D., Kodom K, (2009) Geoelectrical evaluation of soil properties. Geophysical Research Abstracts, Vol. 11, EGU2009-11624-4. Georgescu Paul, (1982), Prospectiuni electrice, Universitatea Bucuresti Loke, M.H. (2004), Tutorial: 2D and 3D electrical imagining survey.